Volume 58, pp. 432-449, 2023.

Rational symbolic cubature rules over the first quadrant in a Cartesian plane

Jilali Abouir, Brahim Benouahmane, and Yassine Chakir

Abstract

In this paper we introduce a new symbolic Gaussian formula for the evaluation of an integral over the first quadrant in a Cartesian plane, in particular with respect to the weight function $w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. It integrates exactly a class of homogeneous Laurent polynomials with coefficients in the commutative field of rational functions in two variables. It is derived using the connection between orthogonal polynomials, two-point Padé approximants, and Gaussian cubatures. We also discuss the connection to two-point Padé-type approximants in order to establish symbolic cubature formulas of interpolatory type. Numerical examples are presented to illustrate the different formulas developed in the paper.

Full Text (PDF) [344 KB], BibTeX , DOI: 10.1553/etna_vol58s432

Key words

homogeneous orthogonal polynomials, homogeneous two-point Padé, symbolic Gaussian cubature

AMS subject classifications

41A21, 41A20, 65D32

Links to the cited ETNA articles

[19]	Vol. 53 (2020), pp. 426-438 Ramón Orive, Juan C. Santos-León, and Miodrag M. Spalević: Cubature formulae for the Gaussian weight. Some old and new rules.